Finite Group Databasebeta

← #28340286

Subgroups — #28340286

Order 13696.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812089C211yesmaximal, characteristic, complemented
2C2×#16812071C211yesmaximal, characteristic
3#20292566C211yesmaximal, characteristic, complemented
4#20292549C211yesmaximal, characteristic, complemented
5#20292540C211yesmaximal, characteristic, complemented
6#20292531C211yesmaximal, characteristic, complemented
7#20292514C211yesmaximal, characteristic, complemented
8#20292505C211yesmaximal, characteristic
9#20292505C211yesmaximal, characteristic
10#20292498C211yesmaximal, characteristic, complemented
11#20292498C211yesmaximal, characteristic, complemented
12#20292498C211yesmaximal, characteristic, complemented
13#20292484C211yesmaximal, characteristic, complemented
14#95×D107C211yesmaximal, characteristic, complemented
15#398×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812089—24no—
23#16812071—24no—
24C2×C4⋊Dic107C2211yescharacteristic
25C2×C4⋊Dic107C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×C4×Dic107C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C2×C4⋊C4×C107C2211yescharacteristic
47C2×C4⋊C4×C107C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C4⋊C4×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53C23×D107C2311yescharacteristic
54C23×D107C2311yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—22no—
57C2×Dic214—41no—
58C2×D428—22no—
59C2×D428—22no—
60C2×D428Q814yescomplemented
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C2×C4×D107—22no—
68C2×C4×D107—22no—
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C2×C4×D107—22no—
72C2×C4×D107—22no—
73C2×C4×D107—22no—
74C2×C4×D107—41no—
75C2×C4×D107—41no—
76C2×C4×D107—41no—
77C2×C4×D107—41no—
78C2×C4×D107—41no—
79C4⋊Dic107—22no—
80C4⋊Dic107—22no—
81Dic107⋊C4—22no—
82Dic107⋊C4—41no—
83(C2×D107)⋊C4—22no—
84(C2×D107)⋊C4—22no—
85(C2×D107)⋊C4—22no—
86(C2×D107)⋊C4—22no—
87(C2×D107)⋊C4—41no—
88(C2×D107)⋊C4—41no—
89(C2×D107)⋊C4—41no—
90(C2×D107)⋊C4D414yes—
91C4×Dic107—22no—
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96C22×C4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C4⋊C4×C107—22no—
100C4⋊C4×C107—22no—
101C4⋊C4×C107—41no—
102C4⋊C4×C107—41no—
103C23×C107C2411yescharacteristic
104C22×D107—21no—
105C22×D107—21no—
106C22×D107—21no—
107C22×D107—21no—
108C22×D107—21no—
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—21no—
112C22×D107—21no—
113C22×D107—21no—
114C22×D107C2×D412yes—
115C22×D107C2×Q812yes—
116C22×D107C2×Q812yes—
117C22×D107#4112yes—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107C2×D412yes—
137C2×C4×C107C2×Q812yes—
138C2×C4×C107#4112yes—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107C2×D412yes—
156C2×Dic107#4112yes—
157D428—28no—
158C4×D107—44no—
159C4×D107—44no—
160C4×D107—44no—
161C4×D107—44no—
162Dic214—44no—
163C22×C107C22×D411yescharacteristic
164C22×C107C22×Q811yescharacteristic
165C22×C107C2×#4111yescharacteristic
166C22×C107C2×#4111yescharacteristic
167C22×C107C2×#4111yescharacteristic
168C22×C107H(5,𝔽2)11yescharacteristic
169C22×C107H(5,𝔽2)11yescharacteristic
170C2×D107—22no—
171C2×D107—22no—
172C2×D107—22no—
173C2×D107—22no—
174C2×D107—22no—
175C2×D107—22no—
176C2×D107—22no—
177C2×D107—22no—
178C2×D107—22no—
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107C22⋊Q814yes—
184C4×C107—22no—
185C4×C107—22no—
186C4×C107—22no—
187C4×C107—41no—
188C4×C107—41no—
189C4×C107—41no—
190C4×C107—41no—
191Dic107—22no—
192Dic107—22no—
193Dic107—41no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107C2×C22⋊Q811yescharacteristic
198C2×C107#52911yescharacteristic
199C2×C107#53811yescharacteristic
200C2×C107#54611yescharacteristic
201C2×C107#54611yescharacteristic
202C2×C107#55411yescharacteristic
203C2×C107#55411yescharacteristic
204D107—24no—
205D107—42no—
206#2450—1071nomaximal, Sylow
207C107#245011yescharacteristic, complemented, Sylow
208C2×C22⋊Q8—1071no—
209C2×C4×D4—1071no—
210#400—1071no—
211#398D10711yescharacteristic, complemented
212#396—1071no—
213#393—1071no—
214#391—1071no—
215#387—1071no—
216#386—1071no—
217#386—1071no—
218#385—1071no—
219#385—1071no—
220#385—1071no—
221#380—1071no—
222C2×#95—1071no—
223C22×Q8—1071no—
224C22×D4—1071no—
225C23×C4—1071no—
226C23×C4—1071no—
227C23×C4—1071no—
228C23×C4—1071no—
229C22⋊Q8—2144no—
230C4×D4—2144no—
231C2×C4⋊C4—1071no—
232C2×C4⋊C4—1071no—
233C2×C4⋊C4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4C2×D10711yescharacteristic
236C2×C4⋊C4C2×D10711yescharacteristic
237C2×C4⋊C4C2×D10711yescharacteristic
238C2×C4⋊C4C2×D10711yescharacteristic
239C2×C22⋊C4—1071no—
240C2×C22⋊C4—1071no—
241C2×C22⋊C4—1071no—
242C2×C22⋊C4—1071no—
243C2×C22⋊C4—1071no—
244C2×C22⋊C4—1071no—
245C2×C22⋊C4—1071no—
246C2×C22⋊C4—1071no—
247C2×C42—1071no—
248#95—1071no—
249#95—1071no—
250#95—1071no—
251#95—1071no—
252#95—1071no—
253#95—1071no—
254#95—1071no—
255#95—1071no—
256#95—1071no—
257#95C2×D10711yescharacteristic
258#95C2×D10711yescharacteristic
259#95C2×D10711yescharacteristic
260C24—1071no—
261C24—1071no—
262C2×Q8—4281no—
263C2×Q8—2142no—
264C2×Q8—2142no—
265C2×D4—1074no—
266C2×D4—2142no—
267C2×D4—2142no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—1071no—
273C22×C4—1071no—
274C22×C4—4281no—
275C22×C4—4281no—
276C22×C4—4281no—
277C22×C4—4281no—
278C22×C4—4281no—
279C22×C4—2142no—
280C22×C4—2142no—
281C22×C4—2142no—
282C22×C4—2142no—
283C22×C4—2142no—
284C22×C4—2142no—
285C22×C4—2142no—
286C22×C4C22×D10711yescharacteristic
287C22×C4C22×D10711yescharacteristic
288C22×C4C22×D10711yescharacteristic
289C22×C4C22×D10711yescharacteristic
290C22×C4C22×D10711yescharacteristic
291C22×C4C22×D10711yescharacteristic
292C22×C4C22×D10711yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—4281no—
298C4⋊C4—2142no—
299C4⋊C4—2142no—
300C4⋊C4—2142no—
301C22⋊C4—1074no—
302C22⋊C4—4281no—
303C22⋊C4—4281no—
304C22⋊C4—4281no—
305C22⋊C4—2142no—
306C22⋊C4—2142no—
307C22⋊C4—2142no—
308C22⋊C4—2142no—
309C42—2142no—
310C23—1072no—
311C23—1072no—
312C23—1072no—
313C23—1072no—
314C23—2141no—
315C23—2141no—
316C23—2141no—
317C23—2141no—
318C23—2141no—
319C23—2141no—
320C23—2141no—
321C23—2141no—
322C23—2141no—
323C23—2141no—
324C23C23×D10711yescharacteristic
325Q8—4284no—
326D4—2148no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—1072no—
346C2×C4—1072no—
347C2×C4—4284no—
348C2×C4—4284no—
349C2×C4—4284no—
350C2×C4—4284no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4—2141no—
358C2×C4—2141no—
359C2×C4—2141no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4D4×D10712yes—
368C2×C4Q8×D10712yes—
369C2×C4#1313381912yes—
370C22—1074no—
371C22—4281no—
372C22—4281no—
373C22—4281no—
374C22—4281no—
375C22—2142no—
376C22—2142no—
377C22—2142no—
378C22—2142no—
379C22—2142no—
380C22—2142no—
381C22—2142no—
382C22—2142no—
383C22—2142no—
384C22C2×D4×D10711yescharacteristic
385C22C2×Q8×D10711yescharacteristic
386C22#41×D10711yescharacteristic
387C22#41×D10711yescharacteristic
388C22C2×#1313381911yescharacteristic
389C22#1681215411yescharacteristic
390C22#1681215511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4281no—
399C4—4281no—
400C4—4281no—
401C4—4281no—
402C4—2142no—
403C4—2142no—
404C2—4282no—
405C2—2144no—
406C2C22⋊Q8×D10711yescharacteristic
407C2#2029316211yescharacteristic
408C2#2029321311yescharacteristic
409C2#2029326611yescharacteristic
410C2#2029326811yescharacteristic
411C2#2029333311yescharacteristic
412C2#2029333811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2450
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16812089C2semidirect
3. #20292566C2semidirect
4. #20292549C2semidirect
5. #20292540C2semidirect
6. #20292531C2semidirect
7. #20292514C2semidirect
10. #20292498C2semidirect
11. #20292498C2semidirect
12. #20292498C2semidirect
13. #20292484C2semidirect
14. #95×D107C2semidirect
15. #398×C107C2semidirect
60. C2×D428Q8semidirect
207. C107#2450semidirect
211. #398D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168120892
C2×#168120712
#202925662
#202925492
#202925402
#202925312
#202925142
#202925052
#202924982
#202924842
#95×D1072
#398×C1072
#2450107